Solve frac{sqrt{2}-4}{sqrt{5}+4}

$$\frac { \sqrt { 2 } - 4 } { \sqrt { 5 } + 4 }$$

Evaluate

$$\frac{4\sqrt{2}+4\sqrt{5}-\sqrt{10}-16}{11}\approx -0.414650136$$

Solution Steps

Rationalize the denominator of \frac{\sqrt{2}-4}{\sqrt{5}+4} by multiplying numerator and denominator by \sqrt{5}-4.

$$\frac{\left(\sqrt{2}-4\right)\left(\sqrt{5}-4\right)}{\left(\sqrt{5}+4\right)\left(\sqrt{5}-4\right)}$$

Consider \left(\sqrt{5}+4\right)\left(\sqrt{5}-4\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(\sqrt{2}-4\right)\left(\sqrt{5}-4\right)}{\left(\sqrt{5}\right)^{2}-4^{2}}$$

Square \sqrt{5}. Square 4.

$$\frac{\left(\sqrt{2}-4\right)\left(\sqrt{5}-4\right)}{5-16}$$

Subtract 16 from 5 to get -11.

$$\frac{\left(\sqrt{2}-4\right)\left(\sqrt{5}-4\right)}{-11}$$

Apply the distributive property by multiplying each term of \sqrt{2}-4 by each term of \sqrt{5}-4.

$$\frac{\sqrt{2}\sqrt{5}-4\sqrt{2}-4\sqrt{5}+16}{-11}$$

To multiply \sqrt{2} and \sqrt{5}, multiply the numbers under the square root.

$$\frac{\sqrt{10}-4\sqrt{2}-4\sqrt{5}+16}{-11}$$

Multiply both numerator and denominator by -1.

$$\frac{-\sqrt{10}+4\sqrt{2}+4\sqrt{5}-16}{11}$$